Theorem 6.9 [1]} Let \(U\)  be Haar distributed on \(\mathrm {Sp}(2N)\) . Let \({\bf a}=(a_1,\dots , a_h)\)  with \(a_i \in \mathbb {Z}_{\ge 0}\) . Then for \(N\ge \sum _{j=1}^ha_jj\) ,
\(\mathbb {E}\left(\prod _{j=1}^h (\mathrm {Tr} (U^j))^{a_j} \right)=\prod _{j=1}^h (-1)^{(j-1)a_j}g_j( a_j),\) 
